Is it possible to push an object when I touch it by moving another object in Object Mode?
Here a solution using RigidBodies:


add a ground (large cube), make it a passive ridig body 
add rigidbody to the two other cubes sitting on the ground, check the "Rigidbody/Animated" box for the 1st cube 
hit space to play the animation then move the first cube toward the second. Collisions should happen...